Madhukesh J K

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These are questions asked by Madhukesh J K

I'm trying to solve the problem which is published in the research article. The paper is attached. 

This work done with the help of shooting technique in maple 18

1.Is it possible to solve in maple 17

2. How to obtain the triple soln

3. How to implement Shooting technique

Please anyone help me to get this 

 

restart:

with(plots):

eq1 := (1+K)*(diff(f(eta), `$`(eta, 3)))+f(eta)*(diff(f(eta), `$`(eta, 2)))+K*(diff(g(eta), eta))-2*(diff(f(eta), eta))^2-M*(diff(f(eta), eta)) = 0;

(1+K)*(diff(diff(diff(f(eta), eta), eta), eta))+f(eta)*(diff(diff(f(eta), eta), eta))+K*(diff(g(eta), eta))-2*(diff(f(eta), eta))^2-M*(diff(f(eta), eta)) = 0

(1)

eq2 := (1+(1/2)*K)*(diff(g(eta), `$`(eta, 2)))+f(eta)*(diff(g(eta), eta))-3*g(eta)*(diff(f(eta), eta))-2*K*g(eta)-K*(diff(f(eta), `$`(eta, 2))) = 0;

(1+(1/2)*K)*(diff(diff(g(eta), eta), eta))+f(eta)*(diff(g(eta), eta))-3*g(eta)*(diff(f(eta), eta))-2*K*g(eta)-K*(diff(diff(f(eta), eta), eta)) = 0

(2)

eq3 := (1+(4/3)*Rd)*(diff(theta(eta), `$`(eta, 2)))/Pr+f(eta)*(diff(theta(eta), eta))+Nb*(diff(chi(eta), eta))*(diff(theta(eta), eta))+Nt*(diff(theta(eta), eta))^2 = 0;

(1+(4/3)*Rd)*(diff(diff(theta(eta), eta), eta))/Pr+f(eta)*(diff(theta(eta), eta))+Nb*(diff(chi(eta), eta))*(diff(theta(eta), eta))+Nt*(diff(theta(eta), eta))^2 = 0

(3)

eq4 := diff(chi(eta), `$`(eta, 2))+Sc*f(eta)*(diff(chi(eta), eta))+Nt*(diff(theta(eta), `$`(eta, 2)))/Nb = 0;

diff(diff(chi(eta), eta), eta)+Sc*f(eta)*(diff(chi(eta), eta))+Nt*(diff(diff(theta(eta), eta), eta))/Nb = 0

(4)

bcs := f(0) = S, (D(f))(0) = -1, g(0) = -m*((D^2)(f))(0), (D(theta))(0) = -A*(1-theta(0)), chi(0) = 1+delta*(D(chi))(0), (D(f))(10) = 0, g(10) = 0, theta(10) = 0, chi(10) = 0;

f(0) = S, (D(f))(0) = -1, g(0) = -m*(D(f))(0)^2, (D(theta))(0) = -A*(1-theta(0)), chi(0) = 1+delta*(D(chi))(0), (D(f))(10) = 0, g(10) = 0, theta(10) = 0, chi(10) = 0

(5)

params := [K = .1, S = 2.1257, m = .5, Pr = 2, Nb = .3, Nt = .5, Sc = 2, Rd = .5, delta = .1, A = 5];

[K = .1, S = 2.1257, m = .5, Pr = 2, Nb = .3, Nt = .5, Sc = 2, Rd = .5, delta = .1, A = 5]

(6)

sol := dsolve(eval([eq1, eq2, eq3, eq4, bcs], params), numeric, output = listprocedure, maxmesh = 5000)

Error, (in dsolve/numeric/bvp/convertsys) too few boundary conditions: expected 10, got 9

 

``


 

Download Triple.mw

In the below paper 

ODEs are solved by using Matlab Software

Is it possible to solve this in Matlab software?  [I think you mean Maple!--Carl Love]

I have attached the Article

j.physa.2019.123959.pdf


I m Stucked with this step

restart; with(plots)

eq1 := 2*n*4^n*eta^((n+1)*(1/2))*(diff(f(eta), `$`(eta, 2)))^(n-1)*(diff(f(eta), `$`(eta, 3)))+4^n*(n+1)*eta^((n-1)*(1/2))*(diff(f(eta), `$`(eta, 2)))^n+4*f(eta)*(diff(f(eta), `$`(eta, 2)))-4*m*(diff(f(eta), eta))^2+m-2*M*(diff(f(eta), eta)) = 0;

2*n*4^n*eta^((1/2)*n+1/2)*(diff(diff(f(eta), eta), eta))^(n-1)*(diff(diff(diff(f(eta), eta), eta), eta))+4^n*(n+1)*eta^((1/2)*n-1/2)*(diff(diff(f(eta), eta), eta))^n+4*f(eta)*(diff(diff(f(eta), eta), eta))-4*m*(diff(f(eta), eta))^2+m-2*M*(diff(f(eta), eta)) = 0

(1)

eq2 := 2*eta*(diff(theta(eta), `$`(eta, 2)))+2*(diff(theta(eta), eta))+Pr*(f(eta)*(diff(theta(eta), eta))-s*(diff(f(eta), eta))*theta(eta))+Nb*(diff(theta(eta), eta))*(diff(phi(eta), eta))+Nt*(diff(theta(eta), `$`(eta, 2)))^2 = 0;

2*eta*(diff(diff(theta(eta), eta), eta))+2*(diff(theta(eta), eta))+Pr*(f(eta)*(diff(theta(eta), eta))-s*(diff(f(eta), eta))*theta(eta))+Nb*(diff(theta(eta), eta))*(diff(phi(eta), eta))+Nt*(diff(diff(theta(eta), eta), eta))^2 = 0

(2)

eq3 := 2*eta*(diff(phi(eta), `$`(eta, 2)))+2*(diff(phi(eta), eta))+Sc*(f(eta)*(diff(phi(eta), eta))-s*(diff(f(eta), eta))*phi(eta))+Nb*(2*eta*(diff(theta(eta), `$`(eta, 2)))+2*(diff(theta(eta), eta)))/Nt = 0;
``

2*eta*(diff(diff(phi(eta), eta), eta))+2*(diff(phi(eta), eta))+Sc*(f(eta)*(diff(phi(eta), eta))-s*(diff(f(eta), eta))*phi(eta))+Nb*(2*eta*(diff(diff(theta(eta), eta), eta))+2*(diff(theta(eta), eta)))/Nt = 0

(3)

eq4 := 2*eta*(diff(chi(eta), `$`(eta, 2)))+2*(diff(chi(eta), eta))+Lb*(f(eta)*(diff(chi(eta), eta))-s*(diff(f(eta), eta))*chi(eta))-Pe*(2*eta*chi(eta)*(diff(phi(eta), `$`(eta, 2)))+2*chi(eta)*(diff(phi(eta), eta))+2*eta*(diff(chi(eta), `$`(eta, 2)))*(diff(phi(eta), `$`(eta, 2)))) = 0;

2*eta*(diff(diff(chi(eta), eta), eta))+2*(diff(chi(eta), eta))+Lb*(f(eta)*(diff(chi(eta), eta))-s*(diff(f(eta), eta))*chi(eta))-Pe*(2*eta*chi(eta)*(diff(diff(phi(eta), eta), eta))+2*chi(eta)*(diff(phi(eta), eta))+2*eta*(diff(diff(chi(eta), eta), eta))*(diff(diff(phi(eta), eta), eta))) = 0

(4)

bcs := (D(f))(a) = 0, f(a) = 2*s*a*(D(phi))(a)/Sc, theta(a) = 1, phi(a) = 1, chi(a) = 1, (D(f))(10) = 1/2, theta(10) = 0, phi(10) = 0, chi(10) = 0;

(D(f))(a) = 0, f(a) = 2*s*a*(D(phi))(a)/Sc, theta(a) = 1, phi(a) = 1, chi(a) = 1, (D(f))(10) = 1/2, theta(10) = 0, phi(10) = 0, chi(10) = 0

(5)

params := {Lb = .1, M = .1, Nb = .6, Nt = .2, Pe = 5, Pr = 6.2, Sc = .1, a = 0.1e-1, m = 1/3, n = 1, s = .1};

{Lb = .1, M = .1, Nb = .6, Nt = .2, Pe = 5, Pr = 6.2, Sc = .1, a = 0.1e-1, m = 1/3, n = 1, s = .1}

(6)

sol := dsolve(eval([eq1, eq2, eq3, eq4, bcs], params), numeric, output = listprocedure, maxmesh = 1024)

Error, (in dsolve/numeric/bvp/convertsys) unable to convert to an explicit first-order system

 

odeplot(sol, [[eta, f(eta)][eta, phi(eta)], [eta, theta(eta)], [eta, chi(eta)]], eta = eval(c, params) .. 10)

Error, invalid subscript selector

 

NULL


 

Download Pblm2.mw

When I try to solve this problem above error displays

Here is the file
 

Download Main.mw


When i try to solve the problem i got error like this 

Please help me get the solution

restart

with(plots):

eq1 := (2*(eta*(diff(f(eta), `$`(eta, 3)))+diff(f(eta), `$`(eta, 2))))/((1-phi[1])^2.5*(1-phi[2])^2.5)+((1-phi[2])*(1-phi[1]+phi[1]*rho[s1]/rho[f])+phi[2]*rho[s2]/rho[f])*f(eta)*(diff(f(eta), `$`(eta, 2)))-sigma[hnf]*M*(diff(f(eta), eta))/(sigma[f]*(m^2+1)) = 0;

2*(eta*(diff(diff(diff(f(eta), eta), eta), eta))+diff(diff(f(eta), eta), eta))/((1-phi[1])^2.5*(1-phi[2])^2.5)+((1-phi[2])*(1-phi[1]+phi[1]*rho[s1]/rho[f])+phi[2]*rho[s2]/rho[f])*f(eta)*(diff(diff(f(eta), eta), eta))-sigma[hnf]*M*(diff(f(eta), eta))/(sigma[f]*(m^2+1)) = 0

(1)

eq2 := k[hnf]*(eta*(diff(theta(eta), `$`(eta, 2)))+diff(theta(eta), eta))/k[f]+(1/2)*Pr*((1-phi[2])*(1-phi[1]+phi[1]*`ρC`[p][s1]/`ρC`[p][f])+phi[2]*`ρC`[p][s2]/`ρC`[p][f])*f(eta)*(diff(theta(eta), eta)) = 0;

k[hnf]*(eta*(diff(diff(theta(eta), eta), eta))+diff(theta(eta), eta))/k[f]+(1/2)*Pr*((1-phi[2])*(1-phi[1]+phi[1]*`ρC`[p][s1]/`ρC`[p][f])+phi[2]*`ρC`[p][s2]/`ρC`[p][f])*f(eta)*(diff(theta(eta), eta)) = 0

(2)

eq3 := 2*(1-phi[1])^2.5*(1-phi[2])^2.5*(eta*(diff(chi(eta), `$`(eta, 2)))+diff(chi(eta), eta))+Sc*f(eta)*(diff(chi(eta), eta))-beta*Sc*(1+delta*theta(eta))^n*exp^(-E/(1+delta*theta(eta)))*chi(eta) = 0;

2*(1-phi[1])^2.5*(1-phi[2])^2.5*(eta*(diff(diff(chi(eta), eta), eta))+diff(chi(eta), eta))+Sc*f(eta)*(diff(chi(eta), eta))-beta*Sc*(1+delta*theta(eta))^n*exp^(-E/(1+delta*theta(eta)))*chi(eta) = 0

(3)

bcs := f(c) = (1/2)*lambda*c, (D(f))(c) = (1/2)*lambda, theta(c) = 1, chi(c) = 1, (D(f))(10) = (1-lambda)*(1/2), theta(10) = 0, chi(10) = 0;

f(c) = (1/2)*lambda*c, (D(f))(c) = (1/2)*lambda, theta(c) = 1, chi(c) = 1, (D(f))(10) = 1/2-(1/2)*lambda, theta(10) = 0, chi(10) = 0

(4)

params := {E = .5, M = .1, Pr = 6.2, Sc = .5, beta = .5, c = .1, delta = .5, lambda = -1, m = .5, n = .1, k[f] = .6130, k[hnf] = 1.039626683, phi[1] = .1, phi[2] = .1, rho[f] = 997.1, rho[s1] = 5180, rho[s2] = 1800, sigma[f] = 0.5e-1, sigma[hnf] = 0.8888888337e-1, `ρC`[p][f] = 4.1668809*10^6, `ρC`[p][s1] = 3470600, `ρC`[p][s2] = 1290600};

{E = .5, M = .1, Pr = 6.2, Sc = .5, beta = .5, c = .1, delta = .5, lambda = -1, m = .5, n = .1, k[f] = .6130, k[hnf] = 1.039626683, phi[1] = .1, phi[2] = .1, rho[f] = 997.1, rho[s1] = 5180, rho[s2] = 1800, sigma[f] = 0.5e-1, sigma[hnf] = 0.8888888337e-1, `ρC`[p][f] = 4166880.900, `ρC`[p][s1] = 3470600, `ρC`[p][s2] = 1290600}

(5)

ODEs := [eq1, eq2, eq3]

[2*(eta*(diff(diff(diff(f(eta), eta), eta), eta))+diff(diff(f(eta), eta), eta))/((1-phi[1])^2.5*(1-phi[2])^2.5)+((1-phi[2])*(1-phi[1]+phi[1]*rho[s1]/rho[f])+phi[2]*rho[s2]/rho[f])*f(eta)*(diff(diff(f(eta), eta), eta))-sigma[hnf]*M*(diff(f(eta), eta))/(sigma[f]*(m^2+1)) = 0, k[hnf]*(eta*(diff(diff(theta(eta), eta), eta))+diff(theta(eta), eta))/k[f]+(1/2)*Pr*((1-phi[2])*(1-phi[1]+phi[1]*`ρC`[p][s1]/`ρC`[p][f])+phi[2]*`ρC`[p][s2]/`ρC`[p][f])*f(eta)*(diff(theta(eta), eta)) = 0, 2*(1-phi[1])^2.5*(1-phi[2])^2.5*(eta*(diff(diff(chi(eta), eta), eta))+diff(chi(eta), eta))+Sc*f(eta)*(diff(chi(eta), eta))-beta*Sc*(1+delta*theta(eta))^n*exp^(-E/(1+delta*theta(eta)))*chi(eta) = 0]

(6)

``

sol := dsolve(eval([ODES, bcs], params), numeric, output = listprocedure)

Error, (in dsolve/numeric/process_input) system must be entered as a set/list of expressions/equations

 

``


 

Download 1.mw
 

 

 

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